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Exy

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- TL;DR Summary
- Given a stable 2 or 3 body system with each object having different masses and circular orbits around the barycenter, how do I calculate each bodies orbital period?

Hello physics,

While this is about sci-fi worldbuilding, I feel like this belongs on this board more.

CONTEXT:

I have been building a fictional neighborhood for our star and using the formula

to get the orbital period of orbiting bodies around a stationary mass. Some of the systems have been decided by RNG to be binary and trinary starsystems however, and now I am researching the 2 and 3 body problems.

The actual problem (more of a thought experiment) goes as follows:

2 bodies with the masses m

The question is now either how to get the orbital period of each body, or how to get the relation between the radii of each objects orbit based on their mass (and any other needed parameters)

Then the question turns to stable 3 body systems with each orbiting the joint center in a circle. The goal here again are each bodies orbital period with circular orbits, leading to a stable system. Each body may again have different masses

One of the systems features a large star in the center with 2 smaller ones orbiting it on directly opposite sides. I used the above formula for this with the 3rd stars mass simply being counted as belonging to the stationary center, but is this correct? I would imagine this to be dubious, but I simply do not know.

What I do unterstand is that these systems are incredibly unstable in reality, but this is just 2 or 3 bodies of different masses orbiting a joint center without being perturbed or losing energy. My current understanding tells me that the 3 body system would need 2 of the bodies to have identical mass, but maybe they don't.

I have looked around the Internet, but have not found anything that I could use. This seems to be a massive rabbithole.

Maybe someone here knows where to go or how to solve this.

EDIT: I have found the answer for the 2 body problem within the wikipedia page for "Barycenter" regarding the distances, but not the orbital period.

While this is about sci-fi worldbuilding, I feel like this belongs on this board more.

CONTEXT:

I have been building a fictional neighborhood for our star and using the formula

to get the orbital period of orbiting bodies around a stationary mass. Some of the systems have been decided by RNG to be binary and trinary starsystems however, and now I am researching the 2 and 3 body problems.

The actual problem (more of a thought experiment) goes as follows:

2 bodies with the masses m

_{1}and m_{2}with m_{1}≠ m_{2}. With both orbiting the barycenter in circular orbits, the above formula does not give usable results.The question is now either how to get the orbital period of each body, or how to get the relation between the radii of each objects orbit based on their mass (and any other needed parameters)

Then the question turns to stable 3 body systems with each orbiting the joint center in a circle. The goal here again are each bodies orbital period with circular orbits, leading to a stable system. Each body may again have different masses

One of the systems features a large star in the center with 2 smaller ones orbiting it on directly opposite sides. I used the above formula for this with the 3rd stars mass simply being counted as belonging to the stationary center, but is this correct? I would imagine this to be dubious, but I simply do not know.

What I do unterstand is that these systems are incredibly unstable in reality, but this is just 2 or 3 bodies of different masses orbiting a joint center without being perturbed or losing energy. My current understanding tells me that the 3 body system would need 2 of the bodies to have identical mass, but maybe they don't.

I have looked around the Internet, but have not found anything that I could use. This seems to be a massive rabbithole.

Maybe someone here knows where to go or how to solve this.

EDIT: I have found the answer for the 2 body problem within the wikipedia page for "Barycenter" regarding the distances, but not the orbital period.

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